Large deviations for quasi-arithmetically self-normalized random variables
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 1-12.

We introduce a family of convex (concave) functions called sup (inf) of powers, which are used as generator functions for a special type of quasi-arithmetic means. Using these means, we generalize the large deviation result on self-normalized statistics that was obtained in the homogeneous case by [Q.-M. Shao, Self-normalized large deviations. Ann. Probab. 25 (1997) 285-328]. Furthermore, in the homogenous case, we derive the Bahadur exact slope for tests using self-normalized statistics.

DOI : 10.1051/ps/2011112
Classification : 60F10, 62F05
Mots-clés : large deviations, self-normalised statistics, Bahadur exact slope 
@article{PS_2013__17__1_0,
     author = {Aubry, Jean-Marie and Zani, Marguerite},
     title = {Large deviations for quasi-arithmetically self-normalized random variables},
     journal = {ESAIM: Probability and Statistics},
     pages = {1--12},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011112},
     mrnumber = {3002993},
     zbl = {1290.60028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011112/}
}
TY  - JOUR
AU  - Aubry, Jean-Marie
AU  - Zani, Marguerite
TI  - Large deviations for quasi-arithmetically self-normalized random variables
JO  - ESAIM: Probability and Statistics
PY  - 2013
SP  - 1
EP  - 12
VL  - 17
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011112/
DO  - 10.1051/ps/2011112
LA  - en
ID  - PS_2013__17__1_0
ER  - 
%0 Journal Article
%A Aubry, Jean-Marie
%A Zani, Marguerite
%T Large deviations for quasi-arithmetically self-normalized random variables
%J ESAIM: Probability and Statistics
%D 2013
%P 1-12
%V 17
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011112/
%R 10.1051/ps/2011112
%G en
%F PS_2013__17__1_0
Aubry, Jean-Marie; Zani, Marguerite. Large deviations for quasi-arithmetically self-normalized random variables. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 1-12. doi : 10.1051/ps/2011112. http://archive.numdam.org/articles/10.1051/ps/2011112/

[1] R.R. Bahadur, Some limit theorems in statistics. CBMS-NSF Regional Conference Series in Appl. Math. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1971). | MR | Zbl

[2] H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 (1952) 493-507. | MR | Zbl

[3] E. Cuvelier and M. Noirhomme-Fraiture, An approach to stochastic process using quasi-arithmetic means, in Recent advances in stochastic modeling and data analysis, World Scientific (2007) 2-9. | MR

[4] E. Cuvelier and M. Noirhomme-Fraiture, Parametric families of probability distributions for functional data using quasi-arithmetic means with Archimedean generators, in Functional and operatorial statistics, Contrib. Statist. Springer (2008) 127-133. | MR

[5] V.H. De La Peña, T.L. Lai and Q.-M. Shao, Self-normalized processes : Limit theory and statistical applications, in Probab. Appl. (New York). Springer-Verlag, Berlin (2009). | Zbl

[6] A. Dembo and Q.-M. Shao, Self-normalized large deviations in vector spaces, in High dimensional probability (Oberwolfach, 1996), Birkhäuser, Basel. Progr. Probab. 43 (1998) 27-32. | MR | Zbl

[7] A. Dembo and Q.-M. Shao, Large and moderate deviations for Hotelling's T2-statistic. Electron. Comm. Probab. 11 (2006) 149-159. | MR | Zbl

[8] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, 2d ed. (1952). | JFM | MR | Zbl

[9] A. Kolmogoroff, Sur la notion de la moyenne. Rendiconti Accad. d. L. Roma 12 (1930) 388-391. | JFM

[10] T.L. Lai and Q.-M. Shao, Self-normalized limit theorems in probability and statistics, in Asymptotic theory in probability and statistics with applications, Int. Press, Somerville, MA Adv. Lect. Math. (ALM) 2 (2008) 3-43. | MR | Zbl

[11] Y. Nikitin, Asymptotic efficiency of non parametric tests. Cambridge University Press (1995). | MR

[12] M. Nagumo, Über eine Klasse der Mittelwerte. Japan. J. Math. 7 (1930) 71-79. | JFM

[13] E. Porcu, J. Mateu and G. Christakos, Quasi-arithmetic means of covariance functions with potential applications to space-time data. J. Multivar. Anal. 100 (2009) 1830-1844. | MR | Zbl

[14] Q.-M. Shao, Self-normalized large deviations. Ann. Probab. 25 (1997) 285-328. | MR | Zbl

[15] Q.-M. Shao, A note on the self-normalized large deviation. Chinese J. Appl. Probab. Statist. 22 (2006) 358-362. | MR | Zbl

[16] A.V. Tchirina, Large deviations for a class of scale-free statistics under the gamma distribution. J. Math. Sci. 128 (2005) 2640-2655. | Zbl

[17] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Trans. Systems Man Cybernet. 18 (1988) 183-190. | MR | Zbl

Cité par Sources :